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Hambleton, Hugh C. Williams 205 $a1st ed. 2018. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2018. 215 $a1 online resource (xix, 493 pages) $cillustrations 225 1 $aCMS Books in Mathematics, Ouvrages de mathématiques de la SMC,$x1613-5237 311 $a3-030-01402-9 327 $aChapter 1- Cubic fields -- Chapter 2- Cubic ideals and lattices -- Chapter 3- Binary cubic forms -- Chapter 4- Construction of all cubic fields of a fixed fundamental discriminant (Renate Scheidler) -- Chapter 5- Cubic Pell equations -- Chapter 6- The minima of forms and units by approximation -- Chapter 7- Voronoi's theory of continued fractions -- Chapter 8- Relative minima adjacent to 1 in a reduced lattice -- Chapter 9- Parametrization of norm 1 elements of K -- Tables and References -- Author Index -- Symbol Index -- General Index. 330 $aThe objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi?s unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics. . 410 0$aCMS Books in Mathematics, Ouvrages de mathématiques de la SMC,$x1613-5237 606 $aGeometry, Algebraic 606 $aNumber theory 606 $aAlgorithms 606 $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019 606 $aNumber Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M25001 606 $aAlgorithms$3https://scigraph.springernature.com/ontologies/product-market-codes/M14018 615 0$aGeometry, Algebraic. 615 0$aNumber theory. 615 0$aAlgorithms. 615 14$aAlgebraic Geometry. 615 24$aNumber Theory. 615 24$aAlgorithms. 676 $a516.35 700 $aHambleton$b Samuel A$4aut$4http://id.loc.gov/vocabulary/relators/aut$0768227 702 $aWilliams$b Hugh C$4aut$4http://id.loc.gov/vocabulary/relators/aut 906 $aBOOK 912 $a9910300110003321 996 $aCubic Fields with Geometry$92050628 997 $aUNINA